Simple fit

In this example we consider the inversion problem of the mid-transit times for a three-planet system. We show the standard way for using nauyaca in order to estimate the planet masses and orbits. This is a synthetic planetary system whose transit ephemeris have been calculated as described in Canul et al. (2021). Unlike the two-planet system considered in the Documentation, here we exemplify the usage of the minimum code necessary to invert the problem. We obtain finally the MCMC posteriors and make a pair of figures to evaluate our results.

In [1]:
import nauyaca as nau
import matplotlib.pyplot as plt
import numpy as np

The input data is in the directory

In [2]:
directory = "./inputs/"

Defining planet objects: Names, boundaries and TTVs data

In [3]:
# Planet 1
P1 = nau.SetPlanet('Planet-b')
P1.mass = [0.001, 10]   # Explore super-Earth mass range
P1.period = [10.47, 10.48]
P1.ecc = [0.0, 0.1]
P1.inclination = [90,90]  # Constant!
P1.ascending_node = [60, 120]

P1.load_ttvs(directory + "3pl_planet1_ttvs.dat")
In [4]:
# Planet 2
P2 = nau.SetPlanet('Planet-c')
P2.mass = [0.001, 10]  
P2.period = [14.10, 14.11] 
P2.ecc = [0.0, 0.1]
P2.inclination = [85,95]
P2.ascending_node = [60, 120]

P2.load_ttvs(directory + "3pl_planet2_ttvs.dat")
In [5]:
# Planet 3
P3 = nau.SetPlanet('Planet-d')
P3.mass = [0.001, 10]  
P3.period = [23.57, 23.58]
P3.ecc = [0.0, 0.1]
P3.inclination = [90.3,90.3]  # Constant!
P3.ascending_node = [88.5, 88.5] # Constant!

P3.load_ttvs(directory + "3pl_planet3_ttvs.dat")

Creating the Planetary System object

In [6]:
PS = nau.PlanetarySystem("MySystem", mstar=0.522, rstar=0.4422)
PS.add_planets([P1, P2, P3])

# Print a summary of the Planetary System
print(PS)
--> Parameter -mass- of planet -Planet-b- set to lower physical limit: 0.0123032
--> Parameter -ecc- of planet -Planet-b- set to lower physical limit: 1e-06
--> Parameter -mass- of planet -Planet-c- set to lower physical limit: 0.0123032
--> Parameter -ecc- of planet -Planet-c- set to lower physical limit: 1e-06
--> Parameter -mass- of planet -Planet-d- set to lower physical limit: 0.0123032
--> Parameter -ecc- of planet -Planet-d- set to lower physical limit: 1e-06

 =========== Planetary System Summary =========== 
--> Parameter -mass- of planet -Planet-b- set to lower physical limit: 0.0123032
--> Parameter -ecc- of planet -Planet-b- set to lower physical limit: 1e-06
--> Parameter -mass- of planet -Planet-c- set to lower physical limit: 0.0123032
--> Parameter -ecc- of planet -Planet-c- set to lower physical limit: 1e-06
--> Parameter -mass- of planet -Planet-d- set to lower physical limit: 0.0123032
--> Parameter -ecc- of planet -Planet-d- set to lower physical limit: 1e-06

System: MySystem
Mstar: 0.522 Msun |  Rstar: 0.4422 Rsun
Number of planets: 3
Planet information:
------
Planet1: Planet-b
  Boundaries:
    mass: (0.0123032, 10)  [M_earth]
    period: (10.47, 10.48)  [d]
    ecc: (1e-06, 0.1)   
    inclination: (90, 90)  [deg]
    argument: (0.0, 360)  [deg]
    mean_anomaly: (0.0, 360)  [deg]
    ascending_node: (60, 120)  [deg]
  TTVs: True
------
Planet2: Planet-c
  Boundaries:
    mass: (0.0123032, 10)  [M_earth]
    period: (14.1, 14.11)  [d]
    ecc: (1e-06, 0.1)   
    inclination: (85, 95)  [deg]
    argument: (0.0, 360)  [deg]
    mean_anomaly: (0.0, 360)  [deg]
    ascending_node: (60, 120)  [deg]
  TTVs: True
------
Planet3: Planet-d
  Boundaries:
    mass: (0.0123032, 10)  [M_earth]
    period: (23.57, 23.58)  [d]
    ecc: (1e-06, 0.1)   
    inclination: (90.3, 90.3)  [deg]
    argument: (0.0, 360)  [deg]
    mean_anomaly: (0.0, 360)  [deg]
    ascending_node: (88.5, 88.5)  [deg]
  TTVs: True

Simulation attributes: 
Reference epoch of the solutions (t0): 0.0 [JD]
Total time of TTVs data (ftime): 1359.0 [days]
Timestep of the simulations (dt): 0.34874 [days]

From the output above it is seen that the observations span around ~1360 days. Setting the simulation attribute ftime = 500 will discard all the observations greater than 500 days. It is useful when you want to quickly explore the possible solutions without evaluating all the transits, as in the case of this example. If not changed, these are the default simulations attributes calculated by nauyaca. Let's change the final time to exemplify the usage:

In [7]:
# Simulation attributes
# Here, t0 is the reference time of the osculating elements
PS.simulation(t0=0, ftime=500)

Save the Planetary System for further examples in json format

In [8]:
PS.save_json()
--> JSON file ./MySystem.json saved

Running the optimizers.

Let's run the optimizers to build an informed initial guess for the MCMC. Increase the number of solutions and specify an adequate number of cores to run in parallel.

In [9]:
optim = nau.Optimizers(PS, nsols=120, cores=8)

opt_solutions = optim.run()
 =========== OPTIMIZATION ===========

--> Starting date:  2021-08-16 15:17
--> Finding 120 solutions using 8 cores
--> Results will be saved at:
     * ./MySystem_cube.opt (normalized)
     * ./MySystem_phys.opt (physical)
--> Reference time of the solutions: 0 [days]
- - - - - - - - - - - - - - - - - - - - -
Solution  |   chi square from optimizers
- - - - - - - - - - - - - - - - - - - - -
 3 |  346729722.432  -->  624449.039  -->  373721.785
 7 |  304026464.383  -->  183995389.906  -->  140560.172
 2 |  314257803.181  -->  1002867.079  -->  108913.637
 1 |  213637841.023  -->  2065509.063  -->  158532.608
 4 |  418220204.410  -->  22166.197  -->  8928.114
 5 |  298811335.711  -->  2716081.077  -->  316567.898
 6 |  431389099.830  -->  122309874.901  -->  87801613.464
 8 |  889653777.894  -->  1281292.026  -->  19199.064
 13 |  158273363.622  -->  490365.607  -->  78315.261
 9 |  456897724.394  -->  370280.636  -->  140265.742
 14 |  397656483.107  -->  922399.208  -->  591199.157
 10 |  260566694.323  -->  181151.896  -->  4881.841
 16 |  234796812.653  -->  449850.549  -->  67313.017
 11 |  148898864.303  -->  353175.804  -->  53973.620
 12 |  178962139.599  -->  60867595.837  -->  7628284.745
 17 |  516034845.454  -->  423387.940  -->  13343.936
 15 |  384999688.191  -->  191988447.004  -->  521612.242
 18 |  405711108.241  -->  560983.627  -->  200128.925
 20 |  210344260.452  -->  117510.509  -->  7852.728
 19 |  388309290.339  -->  68024582.995  -->  24213175.834
 22 |  145549898.313  -->  281954.405  -->  4957.888
 23 |  133378273.952  -->  5570913.565  -->  85726.864
 21 |  625504212.585  -->  99422.347  -->  3157.475
 24 |  253345625.268  -->  80978.065  -->  7336.384
 25 |  264638882.065  -->  213718.551  -->  51542.953
 27 |  310660032.164  -->  8518696.793  -->  908621.180
 26 |  212232818.647  -->  2342336.070  -->  43712.626
 28 |  430677946.380  -->  257562316.284  -->  185894784.777
 30 |  132493737.877  -->  1507966.295  -->  117161.748
 29 |  423289719.783  -->  247859.784  -->  1454.820
 31 |  51301611.840  -->  69367.194  -->  23610.405
 32 |  174424046.399  -->  5120435.247  -->  75686.150
 33 |  376163374.533  -->  582827.612  -->  358622.329
 34 |  49612272.719  -->  6740820.381  -->  1070811.225
 38 |  319138346.520  -->  19011.271  -->  10373.189
 35 |  409230903.125  -->  525551.875  -->  102216.894
 36 |  164175477.837  -->  508460.901  -->  80489.879
 41 |  358094199.337  -->  72559.967  -->  15272.441
 37 |  142429009.152  -->  375570.561  -->  104733.362
 39 |  399791173.006  -->  155108.583  -->  5027.105
 40 |  100796286.417  -->  2911186.877  -->  675776.472
 46 |  418789928.463  -->  183499.843  -->  18596.245
 48 |  319609136.627  -->  1715402.644  -->  56863.031
 42 |  369590229.425  -->  440506.709  -->  3253.099
 43 |  233630283.999  -->  2307141.476  -->  301502.659
 44 |  294956040.182  -->  127079554.442  -->  95401.156
 45 |  40561133.408  -->  36613.172  -->  2405.455
 47 |  67925493.191  -->  2346714.729  -->  108508.380
 49 |  157546353.572  -->  312182.123  -->  67156.338
 50 |  128561872.357  -->  5105844.548  -->  69987.416
 54 |  272762489.642  -->  604729.256  -->  25282.381
 51 |  254124213.557  -->  20151047.536  -->  582304.418
 52 |  145313542.853  -->  552853.759  -->  182956.616
 53 |  135658760.401  -->  562505.720  -->  105211.244
 55 |  36412612.711  -->  347275.326  -->  50459.186
 56 |  282195145.259  -->  254379.809  -->  12144.910
 59 |  544614069.610  -->  208200.495  -->  88561.843
 61 |  192957905.032  -->  21979.837  -->  11208.754
 58 |  416424985.632  -->  4461296.035  -->  219345.314
 57 |  73201284.348  -->  43578301.596  -->  142320.186
 60 |  62045939.292  -->  1696315.798  -->  773737.829
 62 |  7168972.411  -->  330621.745  -->  76713.875
 65 |  307089870.236  -->  171350.202  -->  29220.137
 63 |  269851010.936  -->  293902.986  -->  266217.100
 64 |  267879608.926  -->  2589212.467  -->  81883.846
 66 |  198313652.509  -->  175293116.665  -->  155043.910
 67 |  122695680.215  -->  2757443.638  -->  962184.320
 68 |  152844160.324  -->  549399.554  -->  62485.247
 69 |  62829474.896  -->  894912.168  -->  46893.634
 72 |  160432381.377  -->  1508781.854  -->  68537.568
 70 |  113852241.340  -->  616797.568  -->  119360.707
 73 |  20427813.492  -->  530709.584  -->  88377.694
 71 |  560391463.861  -->  35345994.529  -->  12815331.180
 76 |  283100345.641  -->  120086.126  -->  90218.208
 74 |  181937549.311  -->  491891.913  -->  28973.372
 82 |  53974919.888  -->  47442.576  -->  47282.749
 77 |  80105199.026  -->  173270.607  -->  40507.094
 75 |  130253754.869  -->  203114.545  -->  2151.125
 78 |  59572472.502  -->  16179326.134  -->  1880427.704
 79 |  99234271.941  -->  532891.025  -->  176015.448
 80 |  198980902.053  -->  299144.279  -->  81486.700
 81 |  107065124.231  -->  6390269.934  -->  222344.654
 83 |  75473980.461  -->  18178.058  -->  2514.744
 90 |  552607499.838  -->  93959.754  -->  68585.054
 87 |  315412348.985  -->  544789.418  -->  246167.931
 84 |  193359725.025  -->  1185337.135  -->  41309.819
 85 |  53386792.595  -->  3482934.027  -->  539779.493
 86 |  341122660.194  -->  667316.714  -->  12248.771
 89 |  98072196.762  -->  4275855.283  -->  87089.925
 88 |  159258577.910  -->  28689.898  -->  4976.262
 91 |  179217844.361  -->  49724707.160  -->  1876083.350
 94 |  136190941.886  -->  777430.880  -->  273704.629
 97 |  438793708.926  -->  51055.013  -->  4432.207
 96 |  182073531.224  -->  155909.551  -->  93717.919
 92 |  187884513.974  -->  398947.979  -->  35052.885
 93 |  503100850.004  -->  422803921.033  -->  27933.518
 95 |  186029729.835  -->  36750543.648  -->  2473857.297
 98 |  144195383.734  -->  64201317.984  -->  176385.834
 99 |  110306473.627  -->  479727.107  -->  117672.458
 100 |  478080117.732  -->  200936.494  -->  98159.303
 102 |  256038662.586  -->  15280476.048  -->  3432422.680
 101 |  220416863.741  -->  94149194.779  -->  20314.593
 104 |  665754887.868  -->  614208799.014  -->  419808049.779
 103 |  103558452.145  -->  279909.404  -->  26620.828
 105 |  261989312.601  -->  139058910.059  -->  25268835.291
 106 |  319150632.131  -->  134455981.300  -->  277784.896
 107 |  399893342.736  -->  91480.759  -->  4162.153
 108 |  327510069.752  -->  3634026.101  -->  2005594.977
 109 |  224287112.446  -->  27139.582  -->  3155.901
 110 |  165413403.460  -->  284563.314  -->  12423.351
 112 |  121235052.016  -->  61122445.535  -->  914715.052
 113 |  10876738.520  -->  128536.809  -->  6833.191
 111 |  309958627.607  -->  174546936.867  -->  161239.546
 114 |  674530468.420  -->  227966.216  -->  39389.739
 115 |  284835491.612  -->  187313.339  -->  119934.728
 117 |  150141417.517  -->  27853511.019  -->  11585339.846
 116 |  472599524.026  -->  344860605.281  -->  277806176.029
 118 |  320020213.088  -->  133550153.043  -->  593210.438
 120 |  116518781.432  -->  775986.710  -->  187118.665
 119 |  170123789.070  -->  66663194.923  -->  415345.761
Elapsed time in optimization: 17.689 minutes

Running the MCMC.

Here, we set the kwarg 'opt_data' with the optimizer results, but many other kwargs must be given in order to internally create initial walkers.

Try increasing the number of iterations, walkers or temperatures

In [11]:
mcmc = nau.MCMC(PS,
                tmax=100,          # Maximum temperature in ladder (see ptemcee documentation)
                itmax=12000,       # Maximum number of iterations      
                intra_steps=100,   # thinning factor
                cores=8,           # Cores to run in parallel
                opt_data=np.genfromtxt('MySystem_cube.opt') ,   # Solutions from optimizers (cube). 
                                                                # It can also be the variable -opt_solutions-
                distribution='Ladder',   # A strategy to initialize walkers
                fbest=0.05,         # A fraction of the best optimizer results
                nwalkers=60,       # Number of walkers
                ntemps=12          # Number of temperatures
                )
mcmc.run()
--> Selected distribution: Ladder
    6 of 120 solutions taken

 =========== PARALLEL-TEMPERING MCMC ===========

--> Starting date:  2021-08-16 20:21
--> Reference epoch of the solutions:  0  [JD]
--> Results will be saved at:  ./MySystem.hdf5
--> MCMC parameters:
      -ntemps: 12
      -nwalkers: 60
      -itmax: 12000
      -intra_steps: 100

--------- Iteration:  100
 Mean tau Temp 0: 0.828
 Accepted swap fraction in Temp 0:  0.049
 Mean acceptance fraction Temp 0:  0.243
 Mean log-likelihood:  -424.284
 Mean log-posterior:   -424.284
 Current log-posterior dispersion:  560.487
 Current MAP:  (0, 3) -211.424
 Saving time: 0.01350 sec
 Elapsed time:  0.4259 min
--------- Iteration:  200
 Mean tau Temp 0: 1.617
 Accepted swap fraction in Temp 0:  0.026
 Mean acceptance fraction Temp 0:  0.32
 Mean log-likelihood:  -90.842
 Mean log-posterior:   -90.842
 Current log-posterior dispersion:  98.647
 Current MAP:  (0, 5) -51.791
 Saving time: 0.01714 sec
 Elapsed time:  0.9258 min
--------- Iteration:  300
 Mean tau Temp 0: 2.391
 Accepted swap fraction in Temp 0:  0.017
 Mean acceptance fraction Temp 0:  0.336
 Mean log-likelihood:  35.286
 Mean log-posterior:   35.286
 Current log-posterior dispersion:  13.419
 Current MAP:  (0, 51) 68.62
 Saving time: 0.01820 sec
 Elapsed time:  1.3836 min
--------- Iteration:  400
 Mean tau Temp 0: 3.166
 Accepted swap fraction in Temp 0:  0.013
 Mean acceptance fraction Temp 0:  0.345
 Mean log-likelihood:  134.9
 Mean log-posterior:   134.9
 Current log-posterior dispersion:  11.051
 Current MAP:  (0, 57) 152.555
 Saving time: 0.01827 sec
 Elapsed time:  1.8368 min
--------- Iteration:  500
 Mean tau Temp 0: 3.927
 Accepted swap fraction in Temp 0:  0.01
 Mean acceptance fraction Temp 0:  0.344
 Mean log-likelihood:  185.45
 Mean log-posterior:   185.45
 Current log-posterior dispersion:  6.561
 Current MAP:  (0, 51) 201.731
 Saving time: 0.02690 sec
 Elapsed time:  2.3197 min
--------- Iteration:  600
 Mean tau Temp 0: 4.683
 Accepted swap fraction in Temp 0:  0.009
 Mean acceptance fraction Temp 0:  0.343
 Mean log-likelihood:  224.662
 Mean log-posterior:   224.662
 Current log-posterior dispersion:  6.293
 Current MAP:  (0, 26) 237.642
 Saving time: 0.02732 sec
 Elapsed time:  2.9396 min
--------- Iteration:  700
 Mean tau Temp 0: 5.448
 Accepted swap fraction in Temp 0:  0.007
 Mean acceptance fraction Temp 0:  0.339
 Mean log-likelihood:  266.886
 Mean log-posterior:   266.886
 Current log-posterior dispersion:  7.871
 Current MAP:  (0, 53) 291.227
 Saving time: 0.02768 sec
 Elapsed time:  3.5024 min
--------- Iteration:  800
 Mean tau Temp 0: 6.207
 Accepted swap fraction in Temp 0:  0.006
 Mean acceptance fraction Temp 0:  0.333
 Mean log-likelihood:  302.02
 Mean log-posterior:   302.02
 Current log-posterior dispersion:  9.935
 Current MAP:  (0, 3) 322.285
 Saving time: 0.02479 sec
 Elapsed time:  4.0547 min
--------- Iteration:  900
 Mean tau Temp 0: 6.946
 Accepted swap fraction in Temp 0:  0.006
 Mean acceptance fraction Temp 0:  0.329
 Mean log-likelihood:  363.12
 Mean log-posterior:   363.12
 Current log-posterior dispersion:  14.69
 Current MAP:  (0, 41) 386.731
 Saving time: 0.02411 sec
 Elapsed time:  4.5376 min
--------- Iteration:  1000
 Mean tau Temp 0: 7.623
 Accepted swap fraction in Temp 0:  0.005
 Mean acceptance fraction Temp 0:  0.331
 Mean log-likelihood:  422.722
 Mean log-posterior:   422.722
 Current log-posterior dispersion:  6.12
 Current MAP:  (0, 41) 435.916
 Saving time: 0.02302 sec
 Elapsed time:  4.9947 min
--------- Iteration:  1100
 Mean tau Temp 0: 8.295
 Accepted swap fraction in Temp 0:  0.005
 Mean acceptance fraction Temp 0:  0.33
 Mean log-likelihood:  451.269
 Mean log-posterior:   451.269
 Current log-posterior dispersion:  6.722
 Current MAP:  (0, 55) 465.666
 Saving time: 0.02319 sec
 Elapsed time:  5.4427 min
--------- Iteration:  1200
 Mean tau Temp 0: 8.98
 Accepted swap fraction in Temp 0:  0.004
 Mean acceptance fraction Temp 0:  0.328
 Mean log-likelihood:  469.186
 Mean log-posterior:   469.186
 Current log-posterior dispersion:  4.712
 Current MAP:  (0, 14) 479.871
 Saving time: 0.02420 sec
 Elapsed time:  5.8919 min
--------- Iteration:  1300
 Mean tau Temp 0: 9.669
 Accepted swap fraction in Temp 0:  0.004
 Mean acceptance fraction Temp 0:  0.324
 Mean log-likelihood:  486.506
 Mean log-posterior:   486.506
 Current log-posterior dispersion:  5.892
 Current MAP:  (0, 14) 497.526
 Saving time: 0.02395 sec
 Elapsed time:  6.3419 min
--------- Iteration:  1400
 Mean tau Temp 0: 10.355
 Accepted swap fraction in Temp 0:  0.004
 Mean acceptance fraction Temp 0:  0.318
 Mean log-likelihood:  497.402
 Mean log-posterior:   497.402
 Current log-posterior dispersion:  7.268
 Current MAP:  (0, 41) 511.691
 Saving time: 0.02399 sec
 Elapsed time:  6.7907 min
--------- Iteration:  1500
 Mean tau Temp 0: 11.041
 Accepted swap fraction in Temp 0:  0.003
 Mean acceptance fraction Temp 0:  0.311
 Mean log-likelihood:  510.543
 Mean log-posterior:   510.543
 Current log-posterior dispersion:  9.358
 Current MAP:  (0, 20) 524.461
 Saving time: 0.02479 sec
 Elapsed time:  7.2338 min
--------- Iteration:  1600
 Mean tau Temp 0: 11.728
 Accepted swap fraction in Temp 0:  0.003
 Mean acceptance fraction Temp 0:  0.305
 Mean log-likelihood:  522.979
 Mean log-posterior:   522.979
 Current log-posterior dispersion:  11.81
 Current MAP:  (0, 36) 537.934
 Saving time: 0.02636 sec
 Elapsed time:  7.6832 min
--------- Iteration:  1700
 Mean tau Temp 0: 12.411
 Accepted swap fraction in Temp 0:  0.003
 Mean acceptance fraction Temp 0:  0.297
 Mean log-likelihood:  532.267
 Mean log-posterior:   532.267
 Current log-posterior dispersion:  12.53
 Current MAP:  (0, 29) 545.766
 Saving time: 0.02713 sec
 Elapsed time:  8.1419 min
--------- Iteration:  1800
 Mean tau Temp 0: 13.087
 Accepted swap fraction in Temp 0:  0.003
 Mean acceptance fraction Temp 0:  0.291
 Mean log-likelihood:  538.749
 Mean log-posterior:   538.749
 Current log-posterior dispersion:  14.325
 Current MAP:  (0, 42) 555.104
 Saving time: 0.02773 sec
 Elapsed time:  8.5945 min
--------- Iteration:  1900
 Mean tau Temp 0: 13.761
 Accepted swap fraction in Temp 0:  0.003
 Mean acceptance fraction Temp 0:  0.286
 Mean log-likelihood:  543.365
 Mean log-posterior:   543.365
 Current log-posterior dispersion:  14.471
 Current MAP:  (0, 20) 554.683
 Saving time: 0.02639 sec
 Elapsed time:  9.044 min
--------- Iteration:  2000
 Mean tau Temp 0: 14.431
 Accepted swap fraction in Temp 0:  0.003
 Mean acceptance fraction Temp 0:  0.282
 Mean log-likelihood:  549.006
 Mean log-posterior:   549.006
 Current log-posterior dispersion:  15.968
 Current MAP:  (0, 10) 564.009
 Saving time: 0.02797 sec
 Elapsed time:  9.4877 min
--------- Iteration:  2100
 Mean tau Temp 0: 15.097
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.278
 Mean log-likelihood:  554.927
 Mean log-posterior:   554.927
 Current log-posterior dispersion:  17.048
 Current MAP:  (0, 20) 571.013
 Saving time: 0.02898 sec
 Elapsed time:  9.9496 min
--------- Iteration:  2200
 Mean tau Temp 0: 15.756
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.274
 Mean log-likelihood:  559.502
 Mean log-posterior:   559.502
 Current log-posterior dispersion:  18.202
 Current MAP:  (0, 37) 573.246
 Saving time: 0.02953 sec
 Elapsed time:  10.4069 min
--------- Iteration:  2300
 Mean tau Temp 0: 16.411
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.27
 Mean log-likelihood:  562.637
 Mean log-posterior:   562.637
 Current log-posterior dispersion:  19.06
 Current MAP:  (0, 14) 577.935
 Saving time: 0.02993 sec
 Elapsed time:  10.8561 min
--------- Iteration:  2400
 Mean tau Temp 0: 17.06
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.266
 Mean log-likelihood:  565.007
 Mean log-posterior:   565.007
 Current log-posterior dispersion:  19.526
 Current MAP:  (0, 14) 578.988
 Saving time: 0.03072 sec
 Elapsed time:  11.3083 min
--------- Iteration:  2500
 Mean tau Temp 0: 17.705
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.263
 Mean log-likelihood:  568.0
 Mean log-posterior:   568.0
 Current log-posterior dispersion:  20.207
 Current MAP:  (0, 33) 582.951
 Saving time: 0.03117 sec
 Elapsed time:  11.7576 min
--------- Iteration:  2600
 Mean tau Temp 0: 18.346
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.26
 Mean log-likelihood:  571.2
 Mean log-posterior:   571.2
 Current log-posterior dispersion:  20.868
 Current MAP:  (0, 20) 583.114
 Saving time: 0.03046 sec
 Elapsed time:  12.2115 min
--------- Iteration:  2700
 Mean tau Temp 0: 18.982
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.256
 Mean log-likelihood:  572.138
 Mean log-posterior:   572.138
 Current log-posterior dispersion:  21.277
 Current MAP:  (0, 27) 586.148
 Saving time: 0.03131 sec
 Elapsed time:  12.6624 min
--------- Iteration:  2800
 Mean tau Temp 0: 19.613
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.253
 Mean log-likelihood:  574.228
 Mean log-posterior:   574.228
 Current log-posterior dispersion:  21.897
 Current MAP:  (0, 40) 590.158
 Saving time: 0.03253 sec
 Elapsed time:  13.1092 min
--------- Iteration:  2900
 Mean tau Temp 0: 20.24
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.249
 Mean log-likelihood:  575.57
 Mean log-posterior:   575.57
 Current log-posterior dispersion:  22.242
 Current MAP:  (0, 25) 591.442
 Saving time: 0.03333 sec
 Elapsed time:  13.5571 min
--------- Iteration:  3000
 Mean tau Temp 0: 20.859
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.246
 Mean log-likelihood:  577.25
 Mean log-posterior:   577.25
 Current log-posterior dispersion:  23.039
 Current MAP:  (0, 25) 591.687
 Saving time: 0.03467 sec
 Elapsed time:  14.0036 min
--------- Iteration:  3100
 Mean tau Temp 0: 21.471
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.243
 Mean log-likelihood:  578.525
 Mean log-posterior:   578.525
 Current log-posterior dispersion:  23.132
 Current MAP:  (0, 25) 591.864
 Saving time: 0.01478 sec
 Elapsed time:  14.4511 min
--------- Iteration:  3200
 Mean tau Temp 0: 22.077
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.239
 Mean log-likelihood:  579.153
 Mean log-posterior:   579.153
 Current log-posterior dispersion:  22.764
 Current MAP:  (0, 2) 593.515
 Saving time: 0.02080 sec
 Elapsed time:  14.8983 min
--------- Iteration:  3300
 Mean tau Temp 0: 22.674
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.236
 Mean log-likelihood:  580.124
 Mean log-posterior:   580.124
 Current log-posterior dispersion:  22.585
 Current MAP:  (0, 29) 594.479
 Saving time: 0.02127 sec
 Elapsed time:  15.3648 min
--------- Iteration:  3400
 Mean tau Temp 0: 23.262
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.233
 Mean log-likelihood:  580.355
 Mean log-posterior:   580.355
 Current log-posterior dispersion:  22.734
 Current MAP:  (0, 5) 596.644
 Saving time: 0.02195 sec
 Elapsed time:  15.8088 min
--------- Iteration:  3500
 Mean tau Temp 0: 23.844
 Accepted swap fraction in Temp 0:  0.001
 Mean acceptance fraction Temp 0:  0.23
 Mean log-likelihood:  581.618
 Mean log-posterior:   581.618
 Current log-posterior dispersion:  23.279
 Current MAP:  (0, 46) 596.941
 Saving time: 0.02218 sec
 Elapsed time:  16.2564 min
--------- Iteration:  3600
 Mean tau Temp 0: 24.416
 Accepted swap fraction in Temp 0:  0.001
 Mean acceptance fraction Temp 0:  0.228
 Mean log-likelihood:  581.77
 Mean log-posterior:   581.77
 Current log-posterior dispersion:  24.7
 Current MAP:  (0, 39) 595.179
 Saving time: 0.02183 sec
 Elapsed time:  16.7083 min
--------- Iteration:  3700
 Mean tau Temp 0: 24.978
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.225
 Mean log-likelihood:  582.799
 Mean log-posterior:   582.799
 Current log-posterior dispersion:  22.96
 Current MAP:  (0, 2) 596.996
 Saving time: 0.02174 sec
 Elapsed time:  17.1579 min
--------- Iteration:  3800
 Mean tau Temp 0: 25.531
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.222
 Mean log-likelihood:  583.644
 Mean log-posterior:   583.644
 Current log-posterior dispersion:  22.174
 Current MAP:  (0, 18) 598.441
 Saving time: 0.02233 sec
 Elapsed time:  17.6063 min
--------- Iteration:  3900
 Mean tau Temp 0: 26.072
 Accepted swap fraction in Temp 0:  0.002
 Mean acceptance fraction Temp 0:  0.219
 Mean log-likelihood:  584.212
 Mean log-posterior:   584.212
 Current log-posterior dispersion:  21.585
 Current MAP:  (0, 18) 596.615
 Saving time: 0.02369 sec
 Elapsed time:  18.0493 min
--------- Iteration:  4000
 Mean tau Temp 0: 26.602
 Accepted swap fraction in Temp 0:  0.003
 Mean acceptance fraction Temp 0:  0.217
 Mean log-likelihood:  584.141
 Mean log-posterior:   584.141
 Current log-posterior dispersion:  21.994
 Current MAP:  (0, 48) 598.283
 Saving time: 0.02302 sec
 Elapsed time:  18.5025 min
--------- Iteration:  4100
 Mean tau Temp 0: 27.123
 Accepted swap fraction in Temp 0:  0.003
 Mean acceptance fraction Temp 0:  0.214
 Mean log-likelihood:  584.948
 Mean log-posterior:   584.948
 Current log-posterior dispersion:  20.052
 Current MAP:  (0, 22) 600.261
 Saving time: 0.02417 sec
 Elapsed time:  18.9665 min
--------- Iteration:  4200
 Mean tau Temp 0: 27.634
 Accepted swap fraction in Temp 0:  0.004
 Mean acceptance fraction Temp 0:  0.212
 Mean log-likelihood:  585.856
 Mean log-posterior:   585.856
 Current log-posterior dispersion:  18.315
 Current MAP:  (0, 48) 598.476
 Saving time: 0.02393 sec
 Elapsed time:  19.4147 min
--------- Iteration:  4300
 Mean tau Temp 0: 28.131
 Accepted swap fraction in Temp 0:  0.004
 Mean acceptance fraction Temp 0:  0.209
 Mean log-likelihood:  586.419
 Mean log-posterior:   586.419
 Current log-posterior dispersion:  18.641
 Current MAP:  (0, 49) 599.397
 Saving time: 0.02471 sec
 Elapsed time:  19.8582 min
--------- Iteration:  4400
 Mean tau Temp 0: 28.619
 Accepted swap fraction in Temp 0:  0.005
 Mean acceptance fraction Temp 0:  0.207
 Mean log-likelihood:  587.017
 Mean log-posterior:   587.017
 Current log-posterior dispersion:  18.695
 Current MAP:  (0, 45) 599.31
 Saving time: 0.02583 sec
 Elapsed time:  20.3084 min
--------- Iteration:  4500
 Mean tau Temp 0: 29.094
 Accepted swap fraction in Temp 0:  0.005
 Mean acceptance fraction Temp 0:  0.205
 Mean log-likelihood:  587.384
 Mean log-posterior:   587.384
 Current log-posterior dispersion:  17.007
 Current MAP:  (0, 45) 599.167
 Saving time: 0.02672 sec
 Elapsed time:  20.7644 min
--------- Iteration:  4600
 Mean tau Temp 0: 29.556
 Accepted swap fraction in Temp 0:  0.006
 Mean acceptance fraction Temp 0:  0.202
 Mean log-likelihood:  586.51
 Mean log-posterior:   586.51
 Current log-posterior dispersion:  18.965
 Current MAP:  (0, 35) 598.95
 Saving time: 0.02613 sec
 Elapsed time:  21.2109 min
--------- Iteration:  4700
 Mean tau Temp 0: 30.005
 Accepted swap fraction in Temp 0:  0.006
 Mean acceptance fraction Temp 0:  0.2
 Mean log-likelihood:  588.343
 Mean log-posterior:   588.343
 Current log-posterior dispersion:  16.142
 Current MAP:  (0, 45) 598.636
 Saving time: 0.02592 sec
 Elapsed time:  21.6597 min
--------- Iteration:  4800
 Mean tau Temp 0: 30.445
 Accepted swap fraction in Temp 0:  0.007
 Mean acceptance fraction Temp 0:  0.198
 Mean log-likelihood:  589.068
 Mean log-posterior:   589.068
 Current log-posterior dispersion:  15.549
 Current MAP:  (0, 11) 600.565
 Saving time: 0.02639 sec
 Elapsed time:  22.1067 min
--------- Iteration:  4900
 Mean tau Temp 0: 30.867
 Accepted swap fraction in Temp 0:  0.007
 Mean acceptance fraction Temp 0:  0.196
 Mean log-likelihood:  586.708
 Mean log-posterior:   586.708
 Current log-posterior dispersion:  18.199
 Current MAP:  (0, 15) 599.879
 Saving time: 0.02792 sec
 Elapsed time:  22.5474 min
--------- Iteration:  5000
 Mean tau Temp 0: 31.279
 Accepted swap fraction in Temp 0:  0.008
 Mean acceptance fraction Temp 0:  0.194
 Mean log-likelihood:  589.708
 Mean log-posterior:   589.708
 Current log-posterior dispersion:  14.762
 Current MAP:  (0, 0) 600.235
 Saving time: 0.02743 sec
 Elapsed time:  23.0036 min
--------- Iteration:  5100
 Mean tau Temp 0: 31.627
 Accepted swap fraction in Temp 0:  0.008
 Mean acceptance fraction Temp 0:  0.193
 Mean log-likelihood:  589.025
 Mean log-posterior:   589.025
 Current log-posterior dispersion:  14.048
 Current MAP:  (0, 11) 600.287
 Saving time: 0.02769 sec
 Elapsed time:  23.4508 min
--------- Iteration:  5200
 Mean tau Temp 0: 31.926
 Accepted swap fraction in Temp 0:  0.009
 Mean acceptance fraction Temp 0:  0.191
 Mean log-likelihood:  587.918
 Mean log-posterior:   587.918
 Current log-posterior dispersion:  16.579
 Current MAP:  (0, 11) 600.401
 Saving time: 0.02902 sec
 Elapsed time:  23.9053 min
--------- Iteration:  5300
 Mean tau Temp 0: 32.182
 Accepted swap fraction in Temp 0:  0.009
 Mean acceptance fraction Temp 0:  0.189
 Mean log-likelihood:  589.301
 Mean log-posterior:   589.301
 Current log-posterior dispersion:  13.706
 Current MAP:  (0, 8) 600.306
 Saving time: 0.02986 sec
 Elapsed time:  24.353 min
--------- Iteration:  5400
 Mean tau Temp 0: 32.397
 Accepted swap fraction in Temp 0:  0.01
 Mean acceptance fraction Temp 0:  0.188
 Mean log-likelihood:  588.664
 Mean log-posterior:   588.664
 Current log-posterior dispersion:  15.006
 Current MAP:  (0, 8) 600.37
 Saving time: 0.03068 sec
 Elapsed time:  24.8017 min
--------- Iteration:  5500
 Mean tau Temp 0: 32.568
 Accepted swap fraction in Temp 0:  0.01
 Mean acceptance fraction Temp 0:  0.186
 Mean log-likelihood:  590.335
 Mean log-posterior:   590.335
 Current log-posterior dispersion:  13.49
 Current MAP:  (0, 31) 600.642
 Saving time: 0.03088 sec
 Elapsed time:  25.2489 min
--------- Iteration:  5600
 Mean tau Temp 0: 32.7
 Accepted swap fraction in Temp 0:  0.011
 Mean acceptance fraction Temp 0:  0.184
 Mean log-likelihood:  589.533
 Mean log-posterior:   589.533
 Current log-posterior dispersion:  15.126
 Current MAP:  (0, 41) 600.695
 Saving time: 0.03050 sec
 Elapsed time:  25.6943 min
--------- Iteration:  5700
 Mean tau Temp 0: 32.793
 Accepted swap fraction in Temp 0:  0.011
 Mean acceptance fraction Temp 0:  0.183
 Mean log-likelihood:  590.742
 Mean log-posterior:   590.742
 Current log-posterior dispersion:  12.701
 Current MAP:  (0, 9) 601.694
 Saving time: 0.03249 sec
 Elapsed time:  26.153 min
--------- Iteration:  5800
 Mean tau Temp 0: 32.851
 Accepted swap fraction in Temp 0:  0.012
 Mean acceptance fraction Temp 0:  0.181
 Mean log-likelihood:  590.754
 Mean log-posterior:   590.754
 Current log-posterior dispersion:  11.793
 Current MAP:  (0, 38) 600.953
 Saving time: 0.03457 sec
 Elapsed time:  26.6203 min
--------- Iteration:  5900
 Mean tau Temp 0: 32.873
 Accepted swap fraction in Temp 0:  0.012
 Mean acceptance fraction Temp 0:  0.18
 Mean log-likelihood:  589.846
 Mean log-posterior:   589.846
 Current log-posterior dispersion:  13.097
 Current MAP:  (0, 54) 601.259
 Saving time: 0.03307 sec
 Elapsed time:  27.0758 min
--------- Iteration:  6000
 Mean tau Temp 0: 32.856
 Accepted swap fraction in Temp 0:  0.013
 Mean acceptance fraction Temp 0:  0.179
 Mean log-likelihood:  591.319
 Mean log-posterior:   591.319
 Current log-posterior dispersion:  12.633
 Current MAP:  (0, 5) 600.654
 Saving time: 0.03463 sec
 Elapsed time:  27.5198 min
--------- Iteration:  6100
 Mean tau Temp 0: 32.802
 Accepted swap fraction in Temp 0:  0.013
 Mean acceptance fraction Temp 0:  0.177
 Mean log-likelihood:  590.659
 Mean log-posterior:   590.659
 Current log-posterior dispersion:  11.756
 Current MAP:  (0, 5) 601.747
 Saving time: 0.01492 sec
 Elapsed time:  27.9693 min
--------- Iteration:  6200
 Mean tau Temp 0: 32.713
 Accepted swap fraction in Temp 0:  0.013
 Mean acceptance fraction Temp 0:  0.176
 Mean log-likelihood:  590.907
 Mean log-posterior:   590.907
 Current log-posterior dispersion:  12.053
 Current MAP:  (0, 6) 601.996
 Saving time: 0.02042 sec
 Elapsed time:  28.4172 min
--------- Iteration:  6300
 Mean tau Temp 0: 32.584
 Accepted swap fraction in Temp 0:  0.014
 Mean acceptance fraction Temp 0:  0.175
 Mean log-likelihood:  591.352
 Mean log-posterior:   591.352
 Current log-posterior dispersion:  12.005
 Current MAP:  (0, 31) 601.244
 Saving time: 0.02099 sec
 Elapsed time:  28.8633 min
--------- Iteration:  6400
 Mean tau Temp 0: 32.415
 Accepted swap fraction in Temp 0:  0.014
 Mean acceptance fraction Temp 0:  0.173
 Mean log-likelihood:  591.46
 Mean log-posterior:   591.46
 Current log-posterior dispersion:  11.586
 Current MAP:  (0, 56) 599.849
 Saving time: 0.02218 sec
 Elapsed time:  29.3074 min
--------- Iteration:  6500
 Mean tau Temp 0: 32.206
 Accepted swap fraction in Temp 0:  0.015
 Mean acceptance fraction Temp 0:  0.172
 Mean log-likelihood:  590.341
 Mean log-posterior:   590.341
 Current log-posterior dispersion:  11.947
 Current MAP:  (0, 7) 600.17
 Saving time: 0.02193 sec
 Elapsed time:  29.7552 min
--------- Iteration:  6600
 Mean tau Temp 0: 31.962
 Accepted swap fraction in Temp 0:  0.015
 Mean acceptance fraction Temp 0:  0.171
 Mean log-likelihood:  590.171
 Mean log-posterior:   590.171
 Current log-posterior dispersion:  13.723
 Current MAP:  (0, 35) 600.731
 Saving time: 0.02171 sec
 Elapsed time:  30.1942 min
--------- Iteration:  6700
 Mean tau Temp 0: 31.672
 Accepted swap fraction in Temp 0:  0.016
 Mean acceptance fraction Temp 0:  0.17
 Mean log-likelihood:  591.619
 Mean log-posterior:   591.619
 Current log-posterior dispersion:  10.94
 Current MAP:  (0, 1) 600.322
 Saving time: 0.02228 sec
 Elapsed time:  30.6384 min
--------- Iteration:  6800
 Mean tau Temp 0: 31.34
 Accepted swap fraction in Temp 0:  0.017
 Mean acceptance fraction Temp 0:  0.168
 Mean log-likelihood:  591.258
 Mean log-posterior:   591.258
 Current log-posterior dispersion:  11.574
 Current MAP:  (0, 12) 601.623
 Saving time: 0.02312 sec
 Elapsed time:  31.0837 min
--------- Iteration:  6900
 Mean tau Temp 0: 30.962
 Accepted swap fraction in Temp 0:  0.017
 Mean acceptance fraction Temp 0:  0.167
 Mean log-likelihood:  591.518
 Mean log-posterior:   591.518
 Current log-posterior dispersion:  12.71
 Current MAP:  (0, 57) 601.191
 Saving time: 0.02304 sec
 Elapsed time:  31.5231 min
--------- Iteration:  7000
 Mean tau Temp 0: 30.54
 Accepted swap fraction in Temp 0:  0.018
 Mean acceptance fraction Temp 0:  0.166
 Mean log-likelihood:  591.962
 Mean log-posterior:   591.962
 Current log-posterior dispersion:  11.707
 Current MAP:  (0, 28) 601.51
 Saving time: 0.02415 sec
 Elapsed time:  31.9745 min
--------- Iteration:  7100
 Mean tau Temp 0: 30.073
 Accepted swap fraction in Temp 0:  0.019
 Mean acceptance fraction Temp 0:  0.165
 Mean log-likelihood:  592.833
 Mean log-posterior:   592.833
 Current log-posterior dispersion:  10.781
 Current MAP:  (0, 54) 601.075
 Saving time: 0.02427 sec
 Elapsed time:  32.4229 min
--------- Iteration:  7200
 Mean tau Temp 0: 29.599
 Accepted swap fraction in Temp 0:  0.019
 Mean acceptance fraction Temp 0:  0.164
 Mean log-likelihood:  591.413
 Mean log-posterior:   591.413
 Current log-posterior dispersion:  11.297
 Current MAP:  (0, 35) 602.045
 Saving time: 0.02484 sec
 Elapsed time:  32.8656 min
--------- Iteration:  7300
 Mean tau Temp 0: 29.111
 Accepted swap fraction in Temp 0:  0.02
 Mean acceptance fraction Temp 0:  0.163
 Mean log-likelihood:  591.781
 Mean log-posterior:   591.781
 Current log-posterior dispersion:  12.109
 Current MAP:  (0, 5) 602.121
 Saving time: 0.02491 sec
 Elapsed time:  33.3059 min
--------- Iteration:  7400
 Mean tau Temp 0: 28.615
 Accepted swap fraction in Temp 0:  0.02
 Mean acceptance fraction Temp 0:  0.162
 Mean log-likelihood:  591.568
 Mean log-posterior:   591.568
 Current log-posterior dispersion:  11.77
 Current MAP:  (0, 4) 601.852
 Saving time: 0.02609 sec
 Elapsed time:  33.7411 min
--------- Iteration:  7500
 Mean tau Temp 0: 28.106
 Accepted swap fraction in Temp 0:  0.021
 Mean acceptance fraction Temp 0:  0.161
 Mean log-likelihood:  592.026
 Mean log-posterior:   592.026
 Current log-posterior dispersion:  10.614
 Current MAP:  (0, 54) 602.372
 Saving time: 0.02648 sec
 Elapsed time:  34.18 min
--------- Iteration:  7600
 Mean tau Temp 0: 27.59
 Accepted swap fraction in Temp 0:  0.022
 Mean acceptance fraction Temp 0:  0.16
 Mean log-likelihood:  593.996
 Mean log-posterior:   593.996
 Current log-posterior dispersion:  9.115
 Current MAP:  (0, 10) 603.095
 Saving time: 0.02643 sec
 Elapsed time:  34.6233 min
--------- Iteration:  7700
 Mean tau Temp 0: 27.066
 Accepted swap fraction in Temp 0:  0.023
 Mean acceptance fraction Temp 0:  0.159
 Mean log-likelihood:  593.169
 Mean log-posterior:   593.169
 Current log-posterior dispersion:  9.74
 Current MAP:  (0, 45) 602.512
 Saving time: 0.02750 sec
 Elapsed time:  35.0649 min
--------- Iteration:  7800
 Mean tau Temp 0: 26.535
 Accepted swap fraction in Temp 0:  0.023
 Mean acceptance fraction Temp 0:  0.158
 Mean log-likelihood:  593.017
 Mean log-posterior:   593.017
 Current log-posterior dispersion:  10.49
 Current MAP:  (0, 50) 602.032
 Saving time: 0.02751 sec
 Elapsed time:  35.5018 min
--------- Iteration:  7900
 Mean tau Temp 0: 25.979
 Accepted swap fraction in Temp 0:  0.024
 Mean acceptance fraction Temp 0:  0.157
 Mean log-likelihood:  589.74
 Mean log-posterior:   589.74
 Current log-posterior dispersion:  15.577
 Current MAP:  (0, 6) 602.786
 Saving time: 0.02764 sec
 Elapsed time:  35.9426 min
--------- Iteration:  8000
 Mean tau Temp 0: 25.436
 Accepted swap fraction in Temp 0:  0.025
 Mean acceptance fraction Temp 0:  0.156
 Mean log-likelihood:  592.189
 Mean log-posterior:   592.189
 Current log-posterior dispersion:  12.297
 Current MAP:  (0, 0) 601.927
 Saving time: 0.02883 sec
 Elapsed time:  36.3792 min
--------- Iteration:  8100
 Mean tau Temp 0: 24.877
 Accepted swap fraction in Temp 0:  0.025
 Mean acceptance fraction Temp 0:  0.155
 Mean log-likelihood:  592.583
 Mean log-posterior:   592.583
 Current log-posterior dispersion:  10.815
 Current MAP:  (0, 29) 602.488
 Saving time: 0.02836 sec
 Elapsed time:  36.8166 min
--------- Iteration:  8200
 Mean tau Temp 0: 24.313
 Accepted swap fraction in Temp 0:  0.026
 Mean acceptance fraction Temp 0:  0.154
 Mean log-likelihood:  592.048
 Mean log-posterior:   592.048
 Current log-posterior dispersion:  12.367
 Current MAP:  (0, 3) 601.29
 Saving time: 0.02922 sec
 Elapsed time:  37.2473 min
--------- Iteration:  8300
 Mean tau Temp 0: 23.739
 Accepted swap fraction in Temp 0:  0.027
 Mean acceptance fraction Temp 0:  0.153
 Mean log-likelihood:  592.085
 Mean log-posterior:   592.085
 Current log-posterior dispersion:  11.59
 Current MAP:  (0, 34) 601.439
 Saving time: 0.02921 sec
 Elapsed time:  37.6747 min
--------- Iteration:  8400
 Mean tau Temp 0: 23.164
 Accepted swap fraction in Temp 0:  0.028
 Mean acceptance fraction Temp 0:  0.152
 Mean log-likelihood:  592.376
 Mean log-posterior:   592.376
 Current log-posterior dispersion:  10.486
 Current MAP:  (0, 32) 601.664
 Saving time: 0.02840 sec
 Elapsed time:  38.1054 min
--------- Iteration:  8500
 Mean tau Temp 0: 22.581
 Accepted swap fraction in Temp 0:  0.029
 Mean acceptance fraction Temp 0:  0.151
 Mean log-likelihood:  591.951
 Mean log-posterior:   591.951
 Current log-posterior dispersion:  11.883
 Current MAP:  (0, 26) 602.54
 Saving time: 0.02946 sec
 Elapsed time:  38.5357 min
--------- Iteration:  8600
 Mean tau Temp 0: 21.988
 Accepted swap fraction in Temp 0:  0.03
 Mean acceptance fraction Temp 0:  0.15
 Mean log-likelihood:  593.261
 Mean log-posterior:   593.261
 Current log-posterior dispersion:  10.121
 Current MAP:  (0, 51) 602.621
 Saving time: 0.03271 sec
 Elapsed time:  38.9637 min
--------- Iteration:  8700
 Mean tau Temp 0: 21.392
 Accepted swap fraction in Temp 0:  0.031
 Mean acceptance fraction Temp 0:  0.149
 Mean log-likelihood:  593.538
 Mean log-posterior:   593.538
 Current log-posterior dispersion:  10.461
 Current MAP:  (0, 32) 600.652
 Saving time: 0.03186 sec
 Elapsed time:  39.3985 min
--------- Iteration:  8800
 Mean tau Temp 0: 20.783
 Accepted swap fraction in Temp 0:  0.032
 Mean acceptance fraction Temp 0:  0.148
 Mean log-likelihood:  592.344
 Mean log-posterior:   592.344
 Current log-posterior dispersion:  11.158
 Current MAP:  (0, 55) 601.414
 Saving time: 0.03170 sec
 Elapsed time:  39.8294 min
--------- Iteration:  8900
 Mean tau Temp 0: 20.177
 Accepted swap fraction in Temp 0:  0.033
 Mean acceptance fraction Temp 0:  0.148
 Mean log-likelihood:  592.685
 Mean log-posterior:   592.685
 Current log-posterior dispersion:  9.778
 Current MAP:  (0, 32) 601.953
 Saving time: 0.03329 sec
 Elapsed time:  40.2593 min
--------- Iteration:  9000
 Mean tau Temp 0: 19.561
 Accepted swap fraction in Temp 0:  0.034
 Mean acceptance fraction Temp 0:  0.147
 Mean log-likelihood:  592.391
 Mean log-posterior:   592.391
 Current log-posterior dispersion:  10.778
 Current MAP:  (0, 24) 602.086
 Saving time: 0.03355 sec
 Elapsed time:  40.6949 min
--------- Iteration:  9100
 Mean tau Temp 0: 18.944
 Accepted swap fraction in Temp 0:  0.035
 Mean acceptance fraction Temp 0:  0.146
 Mean log-likelihood:  593.327
 Mean log-posterior:   593.327
 Current log-posterior dispersion:  10.025
 Current MAP:  (0, 6) 602.909
 Saving time: 0.01602 sec
 Elapsed time:  41.1285 min
--------- Iteration:  9200
 Mean tau Temp 0: 18.309
 Accepted swap fraction in Temp 0:  0.036
 Mean acceptance fraction Temp 0:  0.145
 Mean log-likelihood:  592.182
 Mean log-posterior:   592.182
 Current log-posterior dispersion:  11.7
 Current MAP:  (0, 56) 602.004
 Saving time: 0.01996 sec
 Elapsed time:  41.562 min
--------- Iteration:  9300
 Mean tau Temp 0: 17.69
 Accepted swap fraction in Temp 0:  0.037
 Mean acceptance fraction Temp 0:  0.145
 Mean log-likelihood:  592.618
 Mean log-posterior:   592.618
 Current log-posterior dispersion:  10.368
 Current MAP:  (0, 48) 600.896
 Saving time: 0.02345 sec
 Elapsed time:  41.9963 min
--------- Iteration:  9400
 Mean tau Temp 0: 17.054
 Accepted swap fraction in Temp 0:  0.038
 Mean acceptance fraction Temp 0:  0.144
 Mean log-likelihood:  592.589
 Mean log-posterior:   592.589
 Current log-posterior dispersion:  10.923
 Current MAP:  (0, 51) 602.344
 Saving time: 0.02133 sec
 Elapsed time:  42.4262 min
--------- Iteration:  9500
 Mean tau Temp 0: 16.427
 Accepted swap fraction in Temp 0:  0.039
 Mean acceptance fraction Temp 0:  0.143
 Mean log-likelihood:  593.121
 Mean log-posterior:   593.121
 Current log-posterior dispersion:  10.643
 Current MAP:  (0, 41) 603.223
 Saving time: 0.02192 sec
 Elapsed time:  42.8522 min
--------- Iteration:  9600
 Mean tau Temp 0: 15.791
 Accepted swap fraction in Temp 0:  0.041
 Mean acceptance fraction Temp 0:  0.142
 Mean log-likelihood:  594.587
 Mean log-posterior:   594.587
 Current log-posterior dispersion:  8.59
 Current MAP:  (0, 17) 602.992
 Saving time: 0.02250 sec
 Elapsed time:  43.2849 min
--------- Iteration:  9700
 Mean tau Temp 0: 15.153
 Accepted swap fraction in Temp 0:  0.042
 Mean acceptance fraction Temp 0:  0.142
 Mean log-likelihood:  593.477
 Mean log-posterior:   593.477
 Current log-posterior dispersion:  9.743
 Current MAP:  (0, 13) 602.626
 Saving time: 0.02220 sec
 Elapsed time:  43.7125 min
--------- Iteration:  9800
 Mean tau Temp 0: 14.509
 Accepted swap fraction in Temp 0:  0.043
 Mean acceptance fraction Temp 0:  0.141
 Mean log-likelihood:  592.113
 Mean log-posterior:   592.113
 Current log-posterior dispersion:  9.998
 Current MAP:  (0, 55) 601.271
 Saving time: 0.02181 sec
 Elapsed time:  44.1457 min
--------- Iteration:  9900
 Mean tau Temp 0: 13.854
 Accepted swap fraction in Temp 0:  0.044
 Mean acceptance fraction Temp 0:  0.14
 Mean log-likelihood:  592.492
 Mean log-posterior:   592.492
 Current log-posterior dispersion:  9.567
 Current MAP:  (0, 40) 602.343
 Saving time: 0.02403 sec
 Elapsed time:  44.5739 min
--------- Iteration:  10000
 Mean tau Temp 0: 13.199
 Accepted swap fraction in Temp 0:  0.045
 Mean acceptance fraction Temp 0:  0.14
 Mean log-likelihood:  592.495
 Mean log-posterior:   592.495
 Current log-posterior dispersion:  10.627
 Current MAP:  (0, 6) 603.27
 Saving time: 0.02273 sec
 Elapsed time:  45.0122 min
--------- Iteration:  10100
 Mean tau Temp 0: 12.533
 Accepted swap fraction in Temp 0:  0.046
 Mean acceptance fraction Temp 0:  0.139
 Mean log-likelihood:  591.468
 Mean log-posterior:   591.468
 Current log-posterior dispersion:  11.326
 Current MAP:  (0, 2) 602.391
 Saving time: 0.02415 sec
 Elapsed time:  45.445 min
--------- Iteration:  10200
 Mean tau Temp 0: 11.889
 Accepted swap fraction in Temp 0:  0.048
 Mean acceptance fraction Temp 0:  0.138
 Mean log-likelihood:  592.773
 Mean log-posterior:   592.773
 Current log-posterior dispersion:  9.205
 Current MAP:  (0, 15) 602.257
 Saving time: 0.02389 sec
 Elapsed time:  45.8773 min
--------- Iteration:  10300
 Mean tau Temp 0: 11.237
 Accepted swap fraction in Temp 0:  0.049
 Mean acceptance fraction Temp 0:  0.137
 Mean log-likelihood:  592.169
 Mean log-posterior:   592.169
 Current log-posterior dispersion:  9.896
 Current MAP:  (0, 59) 602.54
 Saving time: 0.02401 sec
 Elapsed time:  46.3144 min
--------- Iteration:  10400
 Mean tau Temp 0: 10.59
 Accepted swap fraction in Temp 0:  0.05
 Mean acceptance fraction Temp 0:  0.137
 Mean log-likelihood:  593.359
 Mean log-posterior:   593.359
 Current log-posterior dispersion:  9.266
 Current MAP:  (0, 53) 603.288
 Saving time: 0.02558 sec
 Elapsed time:  46.7459 min
--------- Iteration:  10500
 Mean tau Temp 0: 9.946
 Accepted swap fraction in Temp 0:  0.051
 Mean acceptance fraction Temp 0:  0.136
 Mean log-likelihood:  592.585
 Mean log-posterior:   592.585
 Current log-posterior dispersion:  10.788
 Current MAP:  (0, 14) 601.746
 Saving time: 0.02638 sec
 Elapsed time:  47.1813 min
--------- Iteration:  10600
 Mean tau Temp 0: 9.303
 Accepted swap fraction in Temp 0:  0.052
 Mean acceptance fraction Temp 0:  0.135
 Mean log-likelihood:  593.065
 Mean log-posterior:   593.065
 Current log-posterior dispersion:  9.159
 Current MAP:  (0, 6) 601.404
 Saving time: 0.02734 sec
 Elapsed time:  47.6251 min
--------- Iteration:  10700
 Mean tau Temp 0: 8.673
 Accepted swap fraction in Temp 0:  0.054
 Mean acceptance fraction Temp 0:  0.135
 Mean log-likelihood:  592.702
 Mean log-posterior:   592.702
 Current log-posterior dispersion:  9.269
 Current MAP:  (0, 30) 601.796
 Saving time: 0.02680 sec
 Elapsed time:  48.0684 min
--------- Iteration:  10800
 Mean tau Temp 0: 8.052
 Accepted swap fraction in Temp 0:  0.055
 Mean acceptance fraction Temp 0:  0.134
 Mean log-likelihood:  594.523
 Mean log-posterior:   594.523
 Current log-posterior dispersion:  7.337
 Current MAP:  (0, 21) 601.586
 Saving time: 0.02729 sec
 Elapsed time:  48.5059 min
--------- Iteration:  10900
 Mean tau Temp 0: 7.438
 Accepted swap fraction in Temp 0:  0.056
 Mean acceptance fraction Temp 0:  0.134
 Mean log-likelihood:  593.254
 Mean log-posterior:   593.254
 Current log-posterior dispersion:  8.242
 Current MAP:  (0, 55) 602.951
 Saving time: 0.02778 sec
 Elapsed time:  48.94 min
--------- Iteration:  11000
 Mean tau Temp 0: 6.839
 Accepted swap fraction in Temp 0:  0.057
 Mean acceptance fraction Temp 0:  0.133
 Mean log-likelihood:  593.073
 Mean log-posterior:   593.073
 Current log-posterior dispersion:  9.26
 Current MAP:  (0, 20) 602.555
 Saving time: 0.02992 sec
 Elapsed time:  49.382 min
--------- Iteration:  11100
 Mean tau Temp 0: 6.263
 Accepted swap fraction in Temp 0:  0.058
 Mean acceptance fraction Temp 0:  0.132
 Mean log-likelihood:  592.902
 Mean log-posterior:   592.902
 Current log-posterior dispersion:  9.344
 Current MAP:  (0, 31) 603.059
 Saving time: 0.02934 sec
 Elapsed time:  49.8217 min
--------- Iteration:  11200
 Mean tau Temp 0: 5.717
 Accepted swap fraction in Temp 0:  0.059
 Mean acceptance fraction Temp 0:  0.132
 Mean log-likelihood:  593.967
 Mean log-posterior:   593.967
 Current log-posterior dispersion:  8.226
 Current MAP:  (0, 6) 602.107
 Saving time: 0.02902 sec
 Elapsed time:  50.2597 min
--------- Iteration:  11300
 Mean tau Temp 0: 5.212
 Accepted swap fraction in Temp 0:  0.06
 Mean acceptance fraction Temp 0:  0.131
 Mean log-likelihood:  594.13
 Mean log-posterior:   594.13
 Current log-posterior dispersion:  7.664
 Current MAP:  (0, 52) 601.676
 Saving time: 0.03001 sec
 Elapsed time:  50.7011 min
--------- Iteration:  11400
 Mean tau Temp 0: 4.753
 Accepted swap fraction in Temp 0:  0.061
 Mean acceptance fraction Temp 0:  0.131
 Mean log-likelihood:  592.933
 Mean log-posterior:   592.933
 Current log-posterior dispersion:  9.153
 Current MAP:  (0, 34) 601.799
 Saving time: 0.03144 sec
 Elapsed time:  51.1419 min
--------- Iteration:  11500
 Mean tau Temp 0: 4.376
 Accepted swap fraction in Temp 0:  0.062
 Mean acceptance fraction Temp 0:  0.13
 Mean log-likelihood:  593.308
 Mean log-posterior:   593.308
 Current log-posterior dispersion:  7.777
 Current MAP:  (0, 32) 601.047
 Saving time: 0.03209 sec
 Elapsed time:  51.5897 min
--------- Iteration:  11600
 Mean tau Temp 0: 4.123
 Accepted swap fraction in Temp 0:  0.063
 Mean acceptance fraction Temp 0:  0.13
 Mean log-likelihood:  592.397
 Mean log-posterior:   592.397
 Current log-posterior dispersion:  9.015
 Current MAP:  (0, 13) 603.379
 Saving time: 0.03281 sec
 Elapsed time:  52.0291 min
--------- Iteration:  11700
 Mean tau Temp 0: 4.081
 Accepted swap fraction in Temp 0:  0.065
 Mean acceptance fraction Temp 0:  0.129
 Mean log-likelihood:  592.404
 Mean log-posterior:   592.404
 Current log-posterior dispersion:  9.75
 Current MAP:  (0, 0) 603.246
 Saving time: 0.03326 sec
 Elapsed time:  52.4725 min
--------- Iteration:  11800
 Mean tau Temp 0: 4.415
 Accepted swap fraction in Temp 0:  0.066
 Mean acceptance fraction Temp 0:  0.128
 Mean log-likelihood:  593.166
 Mean log-posterior:   593.166
 Current log-posterior dispersion:  8.746
 Current MAP:  (0, 25) 601.985
 Saving time: 0.03375 sec
 Elapsed time:  52.9099 min
--------- Iteration:  11900
 Mean tau Temp 0: 5.639
 Accepted swap fraction in Temp 0:  0.067
 Mean acceptance fraction Temp 0:  0.128
 Mean log-likelihood:  594.105
 Mean log-posterior:   594.105
 Current log-posterior dispersion:  7.68
 Current MAP:  (0, 17) 603.018
 Saving time: 0.03422 sec
 Elapsed time:  53.3422 min
--------- Iteration:  12000
 Mean tau Temp 0: 12.046
 Accepted swap fraction in Temp 0:  0.068
 Mean acceptance fraction Temp 0:  0.127
 Mean log-likelihood:  593.287
 Mean log-posterior:   593.287
 Current log-posterior dispersion:  8.095
 Current MAP:  (0, 29) 603.026
 Saving time: 0.03460 sec
 Elapsed time:  53.7798 min

--> Maximum number of iterations reached in MCMC
--> Best solutions from the ./MySystem.hdf5 will be written at: ./MySystem.best
--> Reference epoch of the solutions:  0  [JD]
--> Iterations performed:  12000
--> Elapsed time in MCMC: 53.7799 minutes
Out[11]:
<ptemcee.sampler.Sampler at 0x7f101ddb0390>

Print a summary of the results and get a dictionary with the posteriors. Let's make first a burnin of the first 20% of the chains. This burnin will be applied over the posteriors.

In [12]:
burn = 0.2
In [13]:
posteriors = nau.utils.mcmc_summary(PS, 
                        hdf5_file=mcmc.hdf5_filename,
                        burnin=burn,
                        get_posteriors=True)
--> Planetary System:  MySystem
    Stellar mass:  0.522
    Stellar radius:  0.4422
    Number of planets:  3
--> Planets:
       Planet 1: Planet-b
       Planet 2: Planet-c
       Planet 3: Planet-d
    
--------------------------
--> MCMC parameters
    Ntemps:  12
    Nwalkers per temperature:  60
    Number of iterations:  12000
    Thining:  100
    Burnin:  0.2
    Chain shape:  (60, 97, 18)
--------------------------
      RESULTS             
--> Results in File:   ./MySystem.hdf5
--> Reference epoch of the solutions:  0  [JD]
--> Best solution in MCMC
      Logl:  603.37856
               mass period ecc inclination argument mean_anomaly ascending_node
      Planet 1: 0.0738   10.4708   0.0431   90.0   256.1075   285.7213   100.8759
      Planet 2: 2.1322   14.105   0.0419   91.1692   216.4375   217.1319   102.3262
      Planet 3: 0.7461   23.5749   0.0191   90.3   355.4471   234.6966   88.5
--------------------------
--> MCMC medians and 1-sigma errors
             mass1      0.06 ^{+0.028}_{-0.022}
           period1      10.47084 ^{+0.000159}_{-0.000101} 
              ecc1      0.042 ^{+0.00358}_{-0.00394}
         argument1      246.2 ^{+7.046}_{-9.86}
     mean_anomaly1      296.43 ^{+10.679}_{-7.727}
   ascending_node1      103.89 ^{+3.301}_{-4.629}
             mass2      2.04 ^{+0.344}_{-0.374}
           period2      14.11 ^{+0.0}_{-0.0}
              ecc2      0.05 ^{+0.008}_{-0.005}
      inclination2      90.62 ^{+0.479}_{-0.941}
         argument2      207.79 ^{+4.314}_{-5.921}
     mean_anomaly2      226.64 ^{+6.707}_{-4.858}
   ascending_node2      106.4 ^{+3.448}_{-3.637}
             mass3      0.72 ^{+0.18}_{-0.17}
           period3      23.57 ^{+0.0}_{-0.001}
              ecc3      0.04 ^{+0.011}_{-0.015}
         argument3      355.49 ^{+2.629}_{-4.424}
     mean_anomaly3      231.54 ^{+4.282}_{-3.316}

Plot a pair of results. Let's show the masses space for these three planets, where the colors are related to the iterations progress

In [14]:
# See the available keys in posteriors
print("\nAvailable posteriors are:")
print(list(posteriors.keys()))
print()

post1 = 'mass1'
post2 = 'mass2'
post3 = 'mass3'

# Colors show the movement forward
lburn = mcmc.itmax*burn
nsave = int(len(posteriors[post1])/mcmc.nwalkers)
steps = np.array([[lburn+(i*mcmc.intra_steps)]*mcmc.nwalkers for i in range(nsave)]).flatten()

fig, axes = plt.subplots(1,2, figsize=(12,6))
axes[0].scatter(posteriors[post1], posteriors[post2],c=steps, 
                edgecolor='w',cmap='GnBu')
axes[0].set_xlabel(post1)
axes[0].set_ylabel(post2)

im=axes[1].scatter(posteriors[post2], posteriors[post3],c=steps, 
                   edgecolor='w',cmap='GnBu')
axes[1].set_xlabel(post2)
axes[1].set_ylabel(post3)
plt.colorbar(im).set_label('Iteration')

plt.savefig(f'3pl_{post1}_{post2}_{post3}.jpg')
Available posteriors are:
['mass1', 'period1', 'ecc1', 'argument1', 'mean_anomaly1', 'ascending_node1', 'mass2', 'period2', 'ecc2', 'inclination2', 'argument2', 'mean_anomaly2', 'ascending_node2', 'mass3', 'period3', 'ecc3', 'argument3', 'mean_anomaly3']

Plot a pair of built-in figures from the nauyaca's Plots module. -burnin- will discard the first 20% of the initial chains

In [15]:
nauplot = nau.Plots(PS, hdf5_file=mcmc.hdf5_filename, burnin=burn, temperature=0)
In [16]:
# Plot the fitted TTVs
nauplot.TTVs(nsols=10, 
             mode='random'  # 'best'
            );

plt.savefig('3pl_TTVs.jpg')
--> plotting random solutions
In [17]:
# Plot a croner plot
nauplot.corner_plot();

plt.savefig('3pl_corner.jpg')
In [18]:
# Plot histograms of the posteriors
nauplot.hist(titles=True);

plt.savefig('3pl_hist.jpg')

Compare your results! The true planetary parameters for this synthetic system are stored in './inputs/trues':

Planet mass period ecc inc argument mean_anomaly asc_node

1    0.07652, 10.4707, 0.0406, 89.809, 292.62, 246.14, 88.3,

2    1.98033, 14.10509, 0.0244, 89.909, 239.76, 191.31, 88.46,

3    0.6705, 23.57467, 0.0306, 90.336, 82.64, 149.24, 88.51

In example 'restarting_mcmc' we'll see how to restart the mcmc to make more iterations and improve the results.


Quitar todo lo que sigue

In [53]:
nauplot.temperature=10
In [19]:
nauplot.trace_plot(plot_means=True);
In [20]:
nauplot.convergence();
--> Performing Gelman-Rubin test
--> Performing Geweke test
In [21]:
nauplot.monitor();